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Home » Scourge Sonic Shocked the Gaming World—Watch How This Scourge Changed Everything! - Abbey Badges

Scourge Sonic Shocked the Gaming World—Watch How This Scourge Changed Everything! - Abbey Badges

📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 \boxed{\left( \frac{4}{25}, \frac{78}{25} \right)}

📌 Find the maximum value of \( (\sec x + \csc x)^2 \) for \( 0 < x < \frac{\pi}{2} \).

📌 Let \( f(x) = (\sec x + \csc x)^2 = \left( \frac{1}{\cos x} + \frac{1}{\sin x} \right)^2 \).

📌 f(x) = \left( \frac{\sin x + \cos x}{\sin x \cos x} \right)^2

📌 Let \( s = \sin x + \cos x \), \( p = \sin x \cos x \). Note:

📌 s^2 = \sin^2 x + \cos^2 x + 2\sin x \cos x = 1 + 2p \Rightarrow p = \frac{s^2 - 1}{2}

📌 f(x) = \left( \frac{s}{p} \right)^2 = \left( \frac{s}{(s^2 - 1)/2} \right)^2 = \left( \frac{2s}{s^2 - 1} \right)^2 = \frac{4s^2}{(s^2 - 1)^2}

📌 Now, for \( x \in (0, \frac{\pi}{2}) \), \( s = \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \), so \( s

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h^2 + 6^2 = 10^2
h^2 + 36 = 100
h^2 = 64
h = \sqrt{64} = 8
Thus, the length of the altitude is \(\boxed{8 \text{ meters}}\).
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V = \frac{s^3}{6\sqrt{2}}
Given \( V = 100 \), we solve for \( s \):
\frac{s^3}{6\sqrt{2}} = 100

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